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Autocorrelation

Measures how similar a signal is to shifted versions of itself.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Signal Processing
Difficulty: Intermediate
Family: Correlation Analysis
Tier: Extended
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 3
Code example: Not ready
Visualization: Ready

Overview

Autocorrelation compares a signal with time-shifted copies to reveal repetition, delay, or periodic patterns.

Core Concepts

signal sampleslag comparisontime-frequency windowcorrelation analysissample windowsignal transformnoise handling

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n log n)

Average

O(n^2)

Worst

O(n^2)

Space

O(n)

When to Use

Use Autocorrelation when you need to shifts a signal against itself.
Use it for pitch detection with inputs like signal.
Use it when sampled data needs filtering, transformation, smoothing, or frequency analysis.
Use it when the Correlation Analysis tradeoff is acceptable for the problem size.

When Not to Use

Avoid it when sample rate, window size, or boundary handling is not defined.
Avoid it when O(n^2) worst-case behavior is too costly for the expected input size.
Avoid it when the required output is not similarity by lag.

What It Does

Shifts a signal against itself
Computes similarity at each lag

Use Cases

Pitch detection
Period finding
Time-series analysis

Inputs

Signal
Lag range

Outputs

Similarity by lag

Key Formulas (1)

Autocorrelation

R_[lag] = ∑(x_[t] · x_{[t + lag]})

R[lag] = sum(x[t] * x[t + lag])

Measures how strongly a signal matches a shifted copy of itself.

Pseudocode

1.Choose a lag value
2.Multiply overlapping signal samples with shifted samples
3.Sum the products
4.Repeat for each lag
5.Inspect peaks in the lag scores

Canonical Example

Input: samples = [1, 2, 3, 2, 1]
Output: transformed or filtered samples
Explanation: Autocorrelation: The algorithm reads local or frequency structure from the sample sequence.

Implementation Notes

Keep the input contract explicit: Signal; Lag range.
Use this step as the implementation spine: Choose a lag value.
Then preserve the next state change: Multiply overlapping signal samples with shifted samples.
Test with simple signals such as constants, impulses, and single-frequency waves.

Common Pitfalls

FFT-based methods can speed correlation for long signals.
Interpreting transformed values without accounting for scale, phase, or edge effects.
Returning similarity by lag without handling empty, invalid, or boundary inputs.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show a signal sliding against itself with peaks at repeated periods.

Step 1 of 5: Choose a lag value

lag-similarity

Notes

Autocorrelation operates in O(n^2) time and O(n) space within the Correlation Analysis family. Key governing relation: R[lag] = sum(x[t] * x[t + lag]). Often compared with Convolution and Fast Fourier Transform.